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Part 5: Identify key features and graph a hyperbola from general form.
Answer the following questions to determine the key features of the hyperbola based on the equation

shown, and then graph it.

4x^2 + 8x – 9y^2 – 36y – 68 = 0

a) Write the standard equation of the hyperbola. (3 points)
b) What is the center of the hyperbola? (1 point)
c) What are the foci of the hyperbola? (2 points)
d) What are the asymptotes of the hyperbola? (2 points)
e) Sketch a graph of the hyperbola and label the center, foci, guiding box, and asymptotes. (4 points)

Sagot :

lcyWeasel547

First things first. I tried and really hope im correct.

A) (x+1)^2/9 – (y+2)^2/4 = 1

B) (-1,-2)

C) (-4.61, -2) and (2.61, -2)

D) y=2/3x-1.33333 and y=-2/3x-2.66666

View image lcyWeasel547
xero099

a) [tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex].

b) (h, k) = (-1, -2).

c) F₁ (x, y) = (-1 - √13, -2), F₂ (x, y) = (-1 + √13, -2).

d) y = ±2 · (x + 1) / 3 - 2.

e) See the image attached below.

How to analyze a hiperbola

In this question we must analyze the general equation of a hyperbola  and derive all the required information by algebraic and graphic means. All this information is derived from the standard equation of the hyperbola, which is presented below:

[tex]\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}} = 1[/tex]   (1)

Where:

  • a - Horizontal axis distance
  • b - Vertical axis distance
  • h, k - Coordinates of the center of the hyperbola.

The foci are located at the following two points: [tex]F_{1} (x, y) = (h - c, k)[/tex], [tex]F_{2} (x,y) = (h + c, k)[/tex].

Where [tex]c = \sqrt{a^{2}+b^{2}}[/tex].

And the asymptotes of the hyperbola are described by the following formulas:

y₁ = b · (x - h)  / a + k   (2)

y₂ = - b · (x - h) / a + k   (3)

a) We transform the general equation into its standard form:

  1. 4 · x² + 8 · x - 9 · y² - 36 · y - 68 = 0    Given
  2. (2 · x)² + 4 · (2 · x) - (3 · y)² - 12 · (3 · y) - 68 = 0   Definition of power/Associative and commutative properties/Definition of multiplication
  3. [(2 · x)²+ 2 · 2 · (2 · x) + 4] - [(3 · y)² + 2 · 6 · (3 · y) + 36] + 36 - 68 - 4 = 0   Compatiblity with addition/Definition of multiplication/Commutative and associative properties/(-1) · a = - a
  4. (2 · x + 2)² - (3 · y + 6)² - 36 = 0   Perfect square trinomial/Definition of addition
  5. 4 · (x + 1)² - 9 · (y + 2)² = 36   Compatibility with addition/Existence of additive inverse/Modulative property
  6. [tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex]   Definition of power/Compatibility with multiplication/Existence of multiplicative inverse/Result

The standard equation of the hiperbola is [tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex]. [tex]\blacksquare[/tex]

b) From the result found in a) we conclude that the center of the hyperbola is (h, k) = (-1, -2).

c) The foci of the hyperbola are located at the following two points:

[tex]c = \sqrt{3^{2}+2^{2}}[/tex]  

[tex]c = \sqrt{13}[/tex]

F₁ (x, y) = (-1 - √13, -2), F₂ (x, y) = (-1 + √13, -2)

The foci of the hyperbola are F₁ (x, y) = (-1 - √13, -2) and F₂ (x, y) = (-1 + √13, -2). [tex]\blacksquare[/tex]

d) And the asymptotes of the hyperbola are presented below:

y₁ = 2 · x / 3   (2)

y₂ = - 2 · x / 3   (3)

The asymptotes of the hyperbola are y₁ = 2 · (x + 1) / 3 - 2 and y₂ = - 2 · (x + 1) / 3 - 2. [tex]\blacksquare[/tex]

e) The graph of the hyperbola and all its characteristics are presented by means of graphing tool. The result is presented in the image attached below.

To learn more on hyperbolas, we kindly invite to check this verified question: https://brainly.com/question/12919612

View image xero099
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